Finite Group Databasebeta

← #4306437

Subgroups — #4306437

Order 512.

2818 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C212yesmaximal
2C2×#2365C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic
4#15158C211yesmaximal, characteristic
5#15151C211yesmaximal, characteristic
6#15145C211yesmaximal, characteristic
7#15143C212yesmaximal
8#15133C212yesmaximal
9#14903C212yesmaximal
10#14880C212yesmaximal
11#14879C212yesmaximal
12#14816C214yesmaximal
13#14804C214yesmaximal
14C4×#395C211yesmaximal, characteristic
15#13821C212yesmaximal
16#13810C211yesmaximal, characteristic
17C4⋊C42C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2212yes—
22#2399—216no—
23C2×#398C2214yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#395C2212yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#395C2211yescharacteristic
32#2365—28no—
33Q8×C4⋊C4—28no—
34C2×#391C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2214yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#384C2212yes—
44C2×#384C2212yes—
45C2×#384C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2214yes—
52C2×C4×C4⋊C4C2214yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×#376C2212yes—
56#1446C2211yescharacteristic
57#1445C2212yes—
58#1445C2212yes—
59#1444C2214yes—
60#1444C2211yescharacteristic
61#1444C2212yes—
62#1444C2214yes—
63#1444C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2211yescharacteristic
68#1441C2212yes—
69#1441C2212yes—
70#1441C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2214yes—
81#1434C2212yes—
82#1434C2214yes—
83#1434C2214yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×Q8C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—24no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—44no—
98C2×C4×Q8—44no—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#398—216no—
133#398—28no—
134#398—28no—
135#398—44no—
136#398—44no—
137#395—28no—
138#395—216no—
139#395—48no—
140#395D418yes—
141#391—28no—
142#389—24no—
143#389—28no—
144#389—24no—
145#389—28no—
146#389—28no—
147#389—216no—
148#387—28no—
149#387—28no—
150#384—28no—
151#384—28no—
152#384—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—216no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—216no—
160C4×C4⋊C4D418yes—
161C4×C4⋊C4Q8116yescomplemented
162#376—28no—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C2×#95C2314yes—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8C2×D414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C4⋊Q8—232no—
223C4×Q8—232no—
224C4×Q8—432no—
225C4×Q8—432no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×Q818yes—
290C2×C4⋊C4C2×Q818yes—
291C2×C4⋊C4#4114yes—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—42no—
312C2×C42—44no—
313C2×C42—42no—
314C2×C42C2×D418yes—
315C2×C42C2×D414yes—
316C2×C42C2×Q818yes—
317C2×C42C2×Q8116yes—
318C2×C42#4118yes—
319C2×C42#4114yes—
320#95—24no—
321#95—28no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—42no—
345#95—44no—
346#95C2×D414yes—
347#95C2×D418yes—
348#95C2×D418yes—
349#95C2×Q8116yes—
350C24C2511yescharacteristic
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—216no—
354C2×Q8—216no—
355C2×Q8—28no—
356C2×Q8—216no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q818yes—
451C22×C4C22×Q818yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q814yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+414yes—
465C22×C42−1+414yes—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—28no—
473C4⋊C4—216no—
474C4⋊C4—416no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—416no—
484C4⋊C4—416no—
485C4⋊C4—84no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C4⋊C4C4⋊Q8116yes—
489C42—28no—
490C42—216no—
491C42—28no—
492C42—216no—
493C42—216no—
494C42—232no—
495C42—48no—
496C42—416no—
497C42—48no—
498C42—44no—
499C23C23×D411yescharacteristic
500C23C23×D411yescharacteristic
501C23C23×Q812yes—
502C23C22×#4112yes—
503C23C22×#4111yescharacteristic
504C23C22×#4111yescharacteristic
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×2−1+412yes—
508C23C2×2−1+412yes—
509C23#58511yescharacteristic
510Q8—232no—
511Q8—432no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4#52514yes—
606C2×C4#52514yes—
607C2×C4C2×C4⋊Q818yes—
608C2×C4C2×C4⋊Q814yes—
609C2×C4#53318yes—
610C2×C4#53318yes—
611C2×C4D4214yes—
612C2×C4#54718yes—
613C2×C4D4×Q818yes—
614C2×C4#55014yes—
615C22C2×#52511yescharacteristic
616C22C2×#52511yescharacteristic
617C22C22×C4⋊Q811yescharacteristic
618C22C2×#53312yes—
619C22C2×#53312yes—
620C22C2×D4211yescharacteristic
621C22C2×#54712yes—
622C22C2×D4×Q812yes—
623C22C2×#55011yescharacteristic
624C22D4×#4111yescharacteristic
625C22#348611yescharacteristic
626C22#348811yescharacteristic
627C22#349212yes—
628C22#349512yes—
629C22#349911yescharacteristic
630C22#350212yes—
631C22#350312yes—
632C22#350312yes—
633C22#350611yescharacteristic
634C22#350912yes—
635C22#351112yes—
636C22#351911yescharacteristic
637C22#352312yes—
638C4—24no—
639C4—24no—
640C4—28no—
641C4—28no—
642C4—28no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—48no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—84no—
652C4—82no—
653C4—81no—
654C4#239918yescomplemented
655C2C2×#239911yescharacteristic
656C2#3574712yes—
657C2#3574911yescharacteristic
658C2#3593612yes—
659C2#3593711yescharacteristic
660C2#3608911yescharacteristic
661C2#3610111yescharacteristic
662C2#3610311yescharacteristic
663C2D4×C4⋊Q811yescharacteristic
664C2#3613312yes—
665C2#3613412yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
161. C4×C4⋊C4Q8semidirect
654. C4#2399semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23992
C2×#23652
C2×Q8×C4⋊C42
#151582
#151512
#151452
#151432
#151332
#149032
#148802
#148792
#148162
#148042
C4×#3952
#138212
#138102
C4⋊C422